MORE IN Applied Mathematics - 3
MU Electronics and Telecom Engineering (Semester 3)
Applied Mathematics - 3
December 2014
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Prove that f(z)=x2-y+2ixy is analytic and find f'(z).
5 M
1 (b) Find the Fourier series expansion for f(x)=|x|, in (-π, π).
5 M
1 (c) Using laplace transform solve the following differential equation with given condition $\dfrac {d^2y}{dt^2} +y=t,$ given taht y(0)=1&y'(0)=0.
5 M
1 (d) If $\overline A = \nabla (xy+ yz+ zx), \ find \ \nabla\cdot \overline A \ and \ \nabla \times \overline A$
5 M

2 (a) $if \ L[J_0 (t) ] = \dfrac {1}{\sqrt{s^2+1}}$ prove $\int^\infty_0 e^{-6t} t J_0 (4t) dt = 3/500$
6 M
2 (b) Find the directional derivative of &straighthi;=x4+y4+z4 at (1,-2,1) in the direction of AB where B is (2,6,-1). Also find the maximum directional of ϕ at (1,-2,1).
6 M
2 (c) Find the Fourier series expansion for f(x)=4-x2, in (0,2) Hence deduce that $\dfrac {\pi^2}{6}=\dfrac {1}{1^2} + \dfrac {1}{2^2}+ \dfrac {1}{3^2} \cdots \ \cdots$
8 M

3 (a) Prove that $J_{1/2}(x) = \sqrt{\dfrac {2}{\pi x}} \sin x$
6 M
3 (b) Using Green's theorm evaluate $int_c (2x^2-y^2)dx + (x^2+y^2) dy where 'c' is the boundary of the surface enclosed by the line x=0, y=0, x=2, y=2. 6 M 3 (c) i) Find Laplace Transform of \[ e^{-\pi} \int^t_c u \sin 3u \ du ii) Find Laplace Transform of \[ \dfrac {d}{dt} \left ( \dfrac {1-\cos 2t}{t} \right ) 8 M 4 (a) Obtain complex form of Fourier series for the function f(x)=sin ax in (-?, ?), where a is not an integer. 6 M 4 (b) Find the analytic function whose imaginary part is \[ v=\dfrac {x}{x^2+y^2} + \cos h \ y\cdot \cos x$
6 M
4 (c) Find inverse Laplace Transform of following $i) \ \log \left [\dfrac {s^2+a^2}{\sqrt{s+b}} \right ] \\ ii) \ \dfrac {1}{s^3 (s-1)}$
8 M

5 (a) Obtain half-range cosine series for f(x)=x(2-x) in 0
6 M
5 (b) Prove that $\overline{F} = \dfrac {\overline r}{r^3}$ is both irrotational and solenoidal.
6 M
5 (c) Show that the function u=sin x cosh y+2 cos x sinh y+ x2-y2+4xy satisfies Laplace's equation and find it corresponding analytic function.
8 M

6 (a) Evaluate by Stoke's theorem $\int_c (xy \ dx + x y^2 \ dy)$ where C is the square in the xy-plane with vertices (1,0), (0,1), (-1,0) and (0,-1).
6 M
6 (b) Find the bilinear transformation, which maps the points z=1,1,? onto the points w=-i, -1, i.
6 M
6 (c) Show that the general solution of $\dfrac {d^2 y }{d x^2} + 4x^2 y=0 \ is \ y=\sqrt{z} [A J_{1/4} (x^2) + B \ J_{-1/4} (x^2)]$ where A and B are constants.
8 M

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